A new encryption task for mathematically gifted students: Encryption arising from patterns

Abstract

The concept of encryption is noteworthy in terms of both familiarizing mathematically gifted students with technological developments and working with mathematically challenging tasks. Once the proper foundations are established, students can begin to formalize encryption and decryption with algebraic formulas. Encryption can be an important resource for developing functional thinking. Based on the given information, this study designed an encryption algorithm through linear patterns that can be presented as a teaching task in classroom environments to students who are learning at elementary school level and explained the implementation process. The task named “Encryption arising from patterns” is considered important in terms of both creating an encryption algorithm and providing content for the development of mathematical patterns and therefore functional thinking. In the task of “Encryption arising from patterns”, the general term of the linear pattern was created by starting from two prime numbers. The numbers corresponding to the first 29 terms of this linear pattern have been calculated. The letters of the alphabet and the terms of the pattern were paired in order. Then, Caeser’s Cipher was applied to the letters in the alphabet. Thus, the numbers corresponding to the key letters were assigned to the letters in plaintext. The letters of plaintext were sent to the receiver along with the numbers corresponding to the key letters and the first three steps of the linear pattern, and the encryption task was completed.

Keywords

• Cryptography
• Mathematical giftedness
• Functional thinking
• Caesar cipher
• Mathematics education

References

1. Amit, M., & Neria, D. (2008). Rising to the challenge: Using generalization in pattern problems to unearth th algebraic skills of talented pre-algebra students. ZDM, 40(1), 111-129.
2. Assmus, D., & Fritzlar, T. (2022). Mathematical creativity and mathematical giftedness in primary school age-An interview study on creating figural patterns. ZDM-Mathematics Education, 54, 113–131.
3. Bauer, C. (2021). Secret history: The story of cryptology. CRC Press.
4. Blanton, M. L., Brizuela, B. M., Gardiner, A., Sawrey, K., & Newman-Owens, A. (2015). A learning trajectory in 6-year-olds’ thinking about generalizing functional relationships. Journal for Research in Mathematics Education, 46(5), 511–558.
5. Blanton, M. L., & Kaput, J. J. (2004). Elementary grade students’ capacity for functional thinking. In M. J. Høines & A. B. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 135–142). PME.
6. Burns, M. (2000). About teaching mathematics (2nd ed.). Math Solutions Publication.
7. Chua, B. L. (2006). Harry potter and the cryptography with matrices. Australian Mathematics Teacher, 62(3), 25-27.
8. Chua, B. L. (2008). Harry potter and the coding of secrets. Mathematics Teaching in the Middle School, 14(2), 114-121.

9. Eraky, A., Leikin, R., & Hadad, B. S. (2022). Relationships between general giftedness, expertise in mathematics, and mathematical creativity that associated with pattern generalization tasks in different representations. Asian Journal for Mathematics Education, 1(1), 36-51.
10. Erdoğan, F., & Erben, T. (2020). An investigation of the measurement estimation strategies used by gifted students. Journal of Computer and Education Research, 8(15), 201-223.
11. Erdogan, F., & Gul, N. (2022). Reflections from the generalization strategies used by gifted students in the growing geometric pattern task. Journal of Gifted Education and Creativity, 9(4), 369-385.
12. Erol, R., & Saygı, E. (2021). The effect of using cryptology on understanding of function concept. International Journal of Contemporary Educational Research, 8(4), 80-90.
13. Guerrero, L., & Rivera A. (2002). Exploration of patterns and recursive functions. In D. S. Mewborn, P. Sztajn, D. Y. White, H. G. Heide, R. L. Bryant, & K. Nooney (Eds.), Proceedings of the Annual Meeting of the North American Chapter of the InternationalGroup for the Psychology of Mathematics Education (24th, Athens, Georgia,October 26-29) (Vol. 1-4, pp. 262-272). PME-NA.
14. Ho, A. M. (2018). Unlocking ideas: Using escape room puzzles in a cryptography classroom. Primus, 28(9), 835-847.
15. Holden, J. (2017). The mathematics of secrets: Cryptography from Caesar ciphers to digital encryption. Princeton University Press.
16. Katrancı, Y., & Özdemir, A. Ş. (2013). Strengthening the subject of modular arithmetic with the help of RSA encryption. KALEM International Journal of Education and Human Sciences, 3(1), 149-186.
17. Kaur, M. (2008). Cryptography as a pedagogical tool. Primus, 18(2), 198–206.
18. Krutetskii, V. A. (1976). The psychology of mathematical abilities in school children. University of Chicago Press.
19. Leikin, R. (2014). Challenging mathematics with multiple solution tasks and mathematical investigations in geometry. In Y. Li, E. A. Silver, & S. Li (Eds.), Transforming mathematics instruction: Multiple approaches and practices (pp. 59–80). Springer.
20. Leikin, R. (2019). Giftedness and high ability in mathematics. In S. Lerman (Ed.), Encyclopedia of mathematics education. 10-page entry. Springer.
21. Leikin, R. (2021). When practice needs more research: the nature and nurture of mathematical giftedness. ZDM-Mathematics Education, 53, 1579–1589.
22. Leikin, R., Leikin, M., Paz-Baruch, N., Waisman, I., & Lev, M. (2017). On the four types of characteristics of süper mathematically gifted students. High Ability Studies, 28(1), 107-125.
23. Leikin, R., & Sriraman, B. (2022). Empirical research on creativity in mathematics (education): From the wastelands of psychology to the current state of the art. ZDM-Mathematics Education, 54(1), 1–17.
24. Miller, S., & Bossomaier, T. (2021). Privacy, encryption and counter-terrorism. In A. Henschke, A. Reed, S. Robbins, & S. Miller (Eds.), Counter-terrorism, ethics, and technology: emerging challenges at the frontiers of counter-terrorism (pp. 139-154). Springer Press.
25. Ministry of National Education. (2018). Mathematics curriculum (Primary and secondary 1, 2, 3, 4, 5, 6, 7, and 8th grades). Ministry of National Education Publ.
26. National Council of Teachers of Mathematics. (2000). Priciples and standarts fo school mathematics. National Council of Teachers of Mathematics.
27. National Council of Teachers of Mathematics. (2016). Providing opportunities for students with exceptional mathematical promise: A position of the national council of teachers of mathematics. National Council of Teachers of Mathematics.
28. OECD. (2021). PISA 2021 creative thinking framework (3rd draft). PISA 2022. https:// www.oecd.org/pisa/ publications/pisa-2021-assessment-and-analytical-framework.htm
29. Orton, A., & Orton, J. (1999). Pattern and the approach to algebra. In A. Orton (Ed.), Pattern in the teaching and learning of mathematics (pp. 104-120). Cassell.
30. Özdemir, A. Ş., & Erdoğan, F. (2011). Teaching factorial and permutation topics through coding activities. The Western Anatolia Journal of Educational Sciences, 2(3), 19-43.
31. Patterson, B. (2021). Analyzing student understanding of cryptography using the SOLO taxonomy. Cryptologia, 45(5), 439-449.
32. Paz-Baruch, N., Leikin, M., & Leikin, R. (2022). Not any gifted is an expert in mathematics and not any expert in mathematics is gifted. Gifted and Talented International, 37(1), 25-41.
33. Santos, A. (2023). Enhancing Caesar’s Cipher. Edição, 9, 1-8.
34. Schliemann, A. D., Carraher, D. W., & Brizuela, B. M. (2012). Algebra in elementary school. In L. Coulangey & J. P. Drouhard (Eds.), Enseignement de l’algèbre élémentaire: Bilan et perspective (pp. 109–124). (Special Issue in Recherches en Didactique des Mathématiques)
35. Sheffield, L. J. (2018). Commentary paper: A reflection on mathematical creativity and giftedness. In F. M. Singer (Ed.), Mathematical creativity and mathematical giftedness (pp. 405-428). Springer International Publishing.
36. Smith, E. (2008). Representational thinking as a framework for introducing functions in the elementary curriculum. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 133–160). Routledge.
37. Souviney, R. J. (1994). Learning to teach mathematics (2nd ed.). Merrill.
38. Stacey, K. (1989). Finding and using patterns in linear generalising problems. Educational Studies in Mathematics, 20(2), 147-164.
39. Steele, D. (2008). Seventh-grade students’ representations for pictorial growth and change problems. ZDM, 40, 55–64.
40. Wu, J., Jen, E., & Gentry, M. (2018). Validating a classroom perception instrument for gifted students in a universitybased residential program. Journal of Advanced Academics, 29(3), 195–215.